Evaluate the following
`(i)intx^(4)e^(5x)dx` `(ii) intx^(4)cos5x dx`

Explanation

`(i)` Let us integrate `e^(5x)` and differentiate `x^(4)` successively w.r.t.`x`
`intx^(4)e^(5x)dx=x^(4)((e^(5x))/(5))-(4x^(3))((e^(5x))/(25))+(12x^(2))((e^(5x))/(125))-(24x)((e^(5x))/(625))+24*(e^(5x))/(3125)+C`
We have stopped at `5th` step as the differential coefficient of `24` w.r.t. `x` is `0` hence all further terms will become zero.
Hence `intx^(4)e^(5x)dx=e^(5x)[(x^(4))/(5)-(4x^(3))/(25)+(12x^(2))/(125)-(24x)/(625)+(24)/(3125)]+C`
`(ii)` Let us integrate `cos5x` sucessively and differentiate `x^(4)` successively w.r.t.`x`
We have
`intx^(4)cos5xdx`
`=x^(4)((sin5x)/(5))-(4x^(3))((-cos5x)/(25))+(12x^(2))((-sin5x)/(125))-(24x)((cos5x)/(625))+24((sin5x)/(3125))+C`
`=(x^(4)sin5x)/(5)+(4x^(3))/(25)cosx5x-(12x^(2)sin5x)/(125)-(24cos5x)/(625)+(24sin5x)/(3125)+C`